After reading a number of solutions and failed, I have come up with my own solution. I believe there could be many different causes for this problem (such as insufficient harddrive space), but here is what worked for me:
MacBook Pro 10.8.5
Iphone 5
"Cannot be saved on this computer" error
1st Step:
Locate the 'backup' folder. Address is: 'Mactintosh HD'/users/"your user name"/library/application support/mobile sync/backup
If you still cannot find it, google iphone backup folder and find it.
Copy all folders inside to somewhere as a 'backup'.
2nd Step:
Delete all folders inside the 'backup' folder.
My Mac asks me for an admin password, type it in and hit entre.
Now this folder should be 'writable' for your itunes.
3rd Step:
Hit backup bottom on itunes, and a new folder will be created in the 'Backup' folder, which means it worked!
4th Step:
Undo step 1, move all your previous folders back to the 'backup' folder.
Last words:
The problem seem to be an automatic lockdown in many of our folders. Itunes cannot write into the folder unless we give the password to unlock it first. This is quite annoying sometimes. If I am correct, you do not have to do step 1 and 4, simply do some actions in the 'backup' folder and make the machine ask for your password.
Hope it helps.